Chaotic cascades for D-branes on singularities

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Franco, S., He, Y., Herzog, C. and Walcher, J. (2004). Chaotic cascades for D-branes on singularities. Paper presented at the Cargese Summer School, Cargèse, France, 7th - 19th 2004.

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arXiv:hep-th/0412207v2 4 Feb 2005

SINGULARITIES

Sebasti´an Franco1_{, Yang-Hui He}2_{, Christopher Herzog}3_{, Johannes Walcher}4

1. Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2. Dept. of Physics and Math/Physics RG, Univ. of Pennsylvania, Philadelphia, PA 19104, USA 3. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 4. Institute for Advanced Study, Princeton, NJ 08540, USA

sfranco@mit.edu,yanghe@physics.upenn.edu,herzog@kitp.ucsb.edu,walcher@ias.edu

Abstract We briefly review our work on the cascading renormalization group flows

for gauge theories on D-branes probing Calabi-Yau singularities. Such RG flows are sometimes chaotic and exhibit duality walls. We construct supergravity solutions dual to logarithmic flows for these theories. We make new observations about a surface of conformal theories and more complicated supergravity solutions.

1. INTRODUCTION

Extending the revolutionary AdS/CFT correspondence [1] beyond the original relation between N = 4 SYM on N D3-branes and Type IIB supergravity (sugra) inAdS5×S5withN units of RR 5-form flux on the

S5is important to understanding realistic strongly coupled field theories such as QCD.

Two standard extensions have been (1) reducing the SUSY to N = 1 by placing the D3-branes transverse to a Calabi-Yau singularity (the dual sugra background becomes AdS5 ×X5, where X5 is some

non-spherical horizon); and (2) breaking conformal invariance and inducing an RG flow, by introducing fractional branes, i.e., D5-branes wrapped over collapsing 2-cycles of the singularity (in the sugra dual, 3-form fluxes are turned on). A fascinating type of RG flow is the duality cascade: Seiberg duality is used to switch to an alternative description whenever infinite coupling is reached. This idea was introduced in [2] for the gauge theory on D-branes probing the conifold.

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2

x

1

x

3

x

2

(A)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t 0 0.5 1 1.5 2 xi

(B)

0 0.125 0.25 0.375 0.5 0.625 0.75

0.4 0.5 0.6 0.7

0 0.125 0.25 0.375 0.5 0.625 0.75

0.2295 0.22955 0.2296 0.22965 0.2297 0.51343 0.51344 0.51345 0.51346 0.51347 0.51348 0.51349 0.5135

0.2295 0.22955 0.2296 0.22965 0.2297

0.22325 0.22335 0.22345 0.22355 0.51104 0.51106 0.51108 0.5111 0.51112 0.51114 0.51116

0.22325 0.22335 0.22345 0.22355

0.22 0.225 0.23 0.235 0.509

0.51 0.511 0.512 0.513

0.5140.22 0.225 0.23 0.235

23(2) 23(1)

231(2)

232(1)

232(3) 23(2) 23(1) 231(3)

2321(2)

2321(3)

232(1)

232(3)

2323(1)

2323(2) 2312(1) 2312(3) 231(2) 231(3) 2313(2) 2313(1)

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(c) (a)

(C)

Figure 1.1

2. CASCADES IN COUPLING SPACE

There is an interesting way to look at cascading RG flows. In a gauge theory described by a quiver with k gauge groups, the inverse squared couplingsxi ≡1/gi2 are positive and define ak-dimensional cone (R+)k.

Inside this cone, the RG flow generates a trajectory dictated by the beta functions and satisfying Pk

i ri/g2i = constant. Each step between

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bouncing in coupling space, one foresees that cascading RG flows will exhibit chaotic behavior.

3. DUALITY WALLS AND FRACTALS

After introducing the notion of a duality cascade, it is natural to wonder whether some supersymmetric extension of the Standard Model, such as the MSSM, can sit at the IR endpoint of a cascade. This question was posed by Matthew Strassler [3]. Generically, while trying to recon-struct such a RG flow, one encounters a UV accumulation point beyond which Seiberg duality cannot proceed. This phenomenon is dubbed a

duality walland has been constructed for gauge theories engineered with D-branes on singularities [4]. Figure 1.1(B) shows the behavior of cou-plings for a cascade with a duality wall for the theory on D-branes over a complex cone over the Zeroth-Hirzebruch surface F0 1.

Postponing the question of a possible UV completion of duality walls, we can study the dependence of its position on initial couplings. Il-lustrating with F0, the result is remarkable and is presented in

Fig-ure 1.1(C). The curve is a fractal, with concave and convex cusps. When-ever we zoom in on a convex cusp, an infinite, self-similar structure of more cusps emerges.

One subtle point which was not emphasized in [5] involves the ex-istence in coupling space of a codimension two surface of conformal theories for F0 and the other del Pezzo quiver gauge theories. If the

number of gauge couplings isn+ 2, then a naive counting of the linearly independentβ-functions constrains only two combinations of gauge cou-plings when the theory is conformal, leaving an n-dimensional surface of conformal theories. This n-dimensional surface is parametrized on the gravity side by the dilaton and the integral of the NSNS B2 form

throughn−1 independent 2-cycles.

The existence of this codimension two surface may well affect the existence and behavior of the duality wall forF0. In [4], it was assumed

that a generic choice of initial couplings would lie on the conformal surface. However, if the initial conditions do not lie on the conformal surface, one expects large coupling constant corrections to the anomalous dimensions, which will in turn affect the strengths of theβ-functions.

4. SUPERGRAVITY DUALS

The main support for the idea of a cascading RG flow in the original case of the conifold comes from a supergravity dual construction. This

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dual reproduces the logarithmic decrease in the effective number of col-ors towards the IR and also matches the beta functions for the gauge couplings.

In [5], analog supergravity solutions were constructed describing log-arithmic cascades for the gauge theories on D-branes probing complex cones over del Pezzo surfaces. The fact that this was possible is re-markable, since they were obtained without knowing the explicit metric. These supergravity solutions are of the general type studied by Gra˜na and Polchinski [6].

The general form of the metric is a warped product of flat four-dimensional Minkowski space and a Calabi-Yau X

ds2 =Z−1/2_η

µνdxµdxν+Z1/2ds2X , (1.1)

The solution also carries 3-form fluxG3 =F3−_gisH3. In order to preserve

N = 1 supersymmetry,G3must be supported only onX, imaginary

self-dual, a (2,1) form and harmonic. Indeed, it is possible to construct a

G3 satisfying all these condition. It has the form

G3 =

n X

I=1

aI(η+idr

r )∧φI (1.2)

where the φI, I = 1. . . n, are a basis of (1,1) forms orthogonal to the

K¨ahler class of the del Pezzo and η = 1₃dψ+σ

. The one-form σ

satisfies dσ = 2ω, with ω the K¨ahler form on dPn, and 0 ≤ ψ <2π is

the angular coordinate on the circle bundle overdPn.

The intersection product between theφIis R

dPnφI∧φJ =−AIJ, where

AIJ is the Cartan matrix for the exceptional Lie algebra En. There is

a different type of fractional brane associated to each φI, given by

D5-branes wrapping the 2-cycle in the del Pezzo Poincar´e dual toφI.

Let us now study the number of D5-branes and D3-branes associated to these solutions

D5-Branes:. The number of D5-branes is given by the Dirac quan-tization of the RR 3-form F3: aJ = 6πα′MJ. Hence, this family of

solutions are dual to cascades in which the number of fractional branes of each type remains constant.

D3-Branes:. Similarly, the effective number of D3-branes is computed fromF5=F5+∗F5 where F5 =d4x∧d(Z−1) and Z is the warp factor

in (1.1). The factor Z satisfies the equation

∇2_XZ =−1

6|H3|

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In [5], |F3|2 _{was assumed to be a function only of the radius, in which}

case

Z(r) = 2·3

4

9−nα

′2_g2

s

ln(r/r0)

r4 +

1 4r4

X

i,j

MIAIJMJ (1.4)

and from Dirac quantization, the number of D3-branes will grow loga-rithmically: N = ₂3_πgsln(r/r0)PI,JMIAIJMJ. However, generically,

Z may depend on other coordinates on the Calabi-Yau cone X.2 The functionZ averaged over the other coordinates may still be logarithmic inr [7].

5. RECENT DEVELOPMENTS

Recently, there has been further progress in the study of quiver theo-ries and their sugra duals. In [8], a-maximization [9] was used to compute the volume of the 5d horizon of the dual of thedP1gauge theory, yielding

an irrational value. This result corrected previous computations in the literature and was obtained by carefully taking into account the global symmetries that are actually preserved by the superpotential. In [5], the duality cascade fordP1 was analyzed using naive R-charges that did not

take into account these global symmetries. A stable elliptical region in coupling space was found with a self-similar logarithmic cascade. Re-doing the analysis with the new R-charges, we find the same elliptical region albeit with a slightly different shape and center.

The 5d horizon for the complex cone over dP1 is called Y2,1 and is

a member of an infinite family of Sasaki-Einstein geometries denoted

Yp,q. They have S2 ×S3 topology. Their metrics were first found, locally, in [10] and then the global properties were analysed in [11]. In [12], their toric description was worked out. Furthermore, the gauge theory duals to the entireYp,q family have been constructed [13]. These developments change profoundly the status of the AdS/CFT, providing an infinite number of field theories with explicit sugra duals.

Acknowledgments

We would like to thank Qudsia Jabeen Ejaz, Ami Hanany, Pavlos Kazakopoulos,

Igor Klebanov and Joe Polchinski for useful discussions. This research is funded

in part by the CTP and the LNS of MIT and by the department of Physics at UPenn. The research of S. F. was supported in part by U.S. DOE Grant #DE-FC02-94ER40818. The research of Y.-H. H. was supported in part by U.S. DOE Grant #DE-FG02-95ER40893 as well as an NSF Focused Research Grant DMS0139799 for “The Geometry of Superstrings”. C. H. was supported in part by the NSF under

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Grant No. PHY99-07949. The research of J. W. was supported by U.S. DOE Grant #DE-FG02-90ER40542. S. F. would like to thank the organizers of Cargese Summer School, where this material was presented.

References

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[2] I. R. Klebanov and M. J. Strassler, “Supergravity and a confin-ing gauge theory: Duality cascades and chiSB-resolution of naked singularities,” JHEP 0008, 052 (2000) [arXiv:hep-th/0007191]. [3] M. J. Strassler, “Duality in Supersymmetric Field Theory

and an Application to Real Particle Physics,” Talk given at International Workshop on Perspectives of Strong Cou-pling Gauge Theories (SCGT 96), Nagoya, Japan. Available at http://www.eken.phys.nagoya-u.ac.jp/Scgt/proc/

[4] S. Franco, A. Hanany, Y. H. He and P. Kazakopoulos, “Duality walls, duality trees and fractional branes,” arXiv:hep-th/0306092.

[5] S. Franco, Y. H. He, C. Herzog and J. Walcher, Phys. Rev. D70, 046006 (2004) [arXiv:hep-th/0402120].

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[7] C. Herzog, Q. J. Ejaz,and I. Klebanov, “Cascading RG Flows for New Sasaki-Einstein Manifolds,” arXiv:hep-th/0412193.

[8] M. Bertolini, F. Bigazzi and A. L. Cotrone, “New checks and sub-tleties for AdS/CFT and a-maximization,” arXiv:hep-th/0411249. [9] K. Intriligator and B. Wecht, “The exact superconformal

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[10] J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, “Super-symmetric AdS(5) solutions of M-theory,” Class. Quant. Grav.21, 4335 (2004) [arXiv:hep-th/0402153].

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